| L(s) = 1 | + 3-s − 3.46·5-s + 3.46·7-s + 9-s − 3.46·15-s + 6·17-s + 4·19-s + 3.46·21-s − 6.92·23-s + 6.99·25-s + 27-s + 3.46·29-s + 3.46·31-s − 11.9·35-s + 6.92·37-s + 6·41-s + 4·43-s − 3.46·45-s + 6.92·47-s + 4.99·49-s + 6·51-s + 3.46·53-s + 4·57-s − 12·59-s − 6.92·61-s + 3.46·63-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.54·5-s + 1.30·7-s + 0.333·9-s − 0.894·15-s + 1.45·17-s + 0.917·19-s + 0.755·21-s − 1.44·23-s + 1.39·25-s + 0.192·27-s + 0.643·29-s + 0.622·31-s − 2.02·35-s + 1.13·37-s + 0.937·41-s + 0.609·43-s − 0.516·45-s + 1.01·47-s + 0.714·49-s + 0.840·51-s + 0.475·53-s + 0.529·57-s − 1.56·59-s − 0.887·61-s + 0.436·63-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724566718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724566718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39521073874020647516114032943, −9.372667454474063667608579411767, −8.214914240670044582931981527415, −7.889096077611923200869152168690, −7.35668311255951904780242918563, −5.79999482200306713163202956010, −4.60778667873856736773533259212, −3.94473219590269394985990184396, −2.82727781584139604585754955534, −1.16462166520986921322379175886,
1.16462166520986921322379175886, 2.82727781584139604585754955534, 3.94473219590269394985990184396, 4.60778667873856736773533259212, 5.79999482200306713163202956010, 7.35668311255951904780242918563, 7.889096077611923200869152168690, 8.214914240670044582931981527415, 9.372667454474063667608579411767, 10.39521073874020647516114032943
